This \(q\)  is sometimes called the “curvature" of \(r\)  (see [1]}). That curvature is however not the same as the Gaussian curvature in case
\(r(z)\)  comes form a metric. A polarized version of \(q\)  is
\(q(z,w)= \frac{1}{2}\big (\frac{\partial r(z)}{\partial z}+\frac{\partial r(w)}{\partial w}-r(z)r(w)\big ),\) 
